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A monthly survey of math news
Football math. "Strategies on Fourth Down, From a Mathematical Point of View," by Virginia Postrel, appeared in the September 12 2002 New York Times. "Given its field position, should a team punt, kick a field goal or go for the first down?" The problem has been analyzed mathematically by David Romer, an economist at Berkeley, using dynamic programming, "a way to calculate the value of actions that have effects far into the future." An example: Your team has a fourth down on the 2-yard line. Punt (and take "the easy 3 points from a field goal") or go for a touchdown? The touchdown, with its almost automatic extra point, has probability 40% in that position; so its expected value, 2.8 points, is less than the expected value of a punt. But, says Romer, the resulting field position for the opposing team must also be considered. He calculates (using data from 700 NFL games) that in a first quarter "the value of a first-and-10 on a team's own 1-yard line ... is minus 1.6 points." Whereas the kickoff after a field goal would leave them at the 27-yard line, on the average, where a first-and-10 is worth +0.6 points (following the algorithm sketched in the Times). This difference brings to 5 points the expected value of the try for touchdown, and makes it it clearly preferable to the punt. Postrel asked Romer if his paper (available from from the National Bureau of Economic Research for $5) could begin to change fourth-down strategies. "If a football coach called me and said, 'I have a new way to deal with the Social Security problem,' ... my first reaction would be, 'They're not qualified to talk about this. Who are they to think that they've solved this?' "
|Probability of a satisfying assignment for the 3-SAT problem and computational cost, plotted against the ratio alpha of constraints to variables. Images from David Mitchell, Bart Selman, Hector Levesque, reproduced with permission. Full-size images available from Bart Selman.|
Phase transitions occur in physical systems, often at a certain "critical temperature" (e.g. ice to and from water at zero degrees C). In "Analytic and Algorithmic Solution of Random Satisfiability Problems" (Science , August 2 2002), Marc Mézard, Giorgio Parisi and Riccardo Zecchina (Orsay) bring methods from statistical mechanics to study a phase transition which occurrs in a purely mathematical context: the probability that a randomly generated k-SAT problem has at least one satisfying ("SAT") assignment. The k means that each constraint involves exactly k variables, so (A+B+c)(a+D+e)(b+E+C)(d+a+b)=1 is a 3-SAT problem with four constraints in the five Boolean variables A,B,C,D,E, with a=(not A), etc. The + is the logical "or": x+y=1 unless x=y=0, and multiplication is the logical "and": xy=0 unless x=y=1. In this example A=0,B=1,C=1,D=0,E=1 is a "satisfying assignment." The role of temperature is played by the ratio alpha of the number of logical constraints to the number of variables. Clearly when there are many more variables than constraints the probability of a satisfying assignment is high, and vice-versa. David Mitchell, Bart Selman and Hector Levesque showed experimentally about 10 years ago that the transition from high to low occurs abruptly at a critical value alphac near 4.3 for k=3 and in addition that the computing time necessary to settle the problem peaks dramatically near alphac. Mézard and his colleagues pin down alphac to 4.256 and locate another transition point alphad = 3.921 such that between alphad and alphac "the space of configurations breaks up into many states, and there exists a nontrivial complexity" thus partly explaining the computation peak observed by Mitchell et al.. They remark "From the strict mathematical point of view, the phase diagram we propose should be considered as a conjecture," an invitation for mathematicians to get involved in this aspect of mathematics.
Ig Noble math. The Ig Nobel prizes are awarded annually at a ceremony described as "the Nobels' madcap cousin" by In-Sung Yoo in the October 7 2002 USA Today. At least two of the prizes rewarded work with a mathematical component: "Arnd Leike of the University of Munich ... won in physics for his work demonstrating how beer froth obeys the mathematical Law of Exponential Decay" (available online), while "the economics award went to the executives and auditors of 28 companies, including Enron, Adelphia, WorldCom and Qwest Communications, for 'adapting the mathematical concept of imaginary numbers for use in the business world.'" And then there was the prize in Mathematics, ignored by USA Today but picked up by the Christian Science Monitor ("Three cheers, and a few jeers, for weird science" by Marius Hentea and Nathan Welton, October 15, 2002): Mathematics: "Estimation of the Total Surface Area in Indian Elephants," K.P. Sreekumar and G. Nirmalan. The prizes are given partly in jest and partly to remind the public that science can actually be fun. The CSM quotes Charles Paxton (St. Andrews) who won this year's biology prize for work on courtship behaviour of ostriches towards humans. "I'm very keen on the popularization of science," Paxton said, "no matter what the personal embarrassment." (articles available online:USA Today CSM.)
The next big thing. The Chronicle of Higher Education (September 30, 2002; Section B, page 4) invited experts in Geography, Math, Information Technologies and Criticism to tell us "What will be the next big thing?" in their fields. The mathematics respondent was John Ewing of the AMS. "The next big thing in mathematics? Biology. ... The mathematics involved in studying the genome and the folding of proteins is deep, elegant, and beautiful ... a spectacular new area of research that is certain to grow enormously in the next 10 years." Ewing goes on: "During the coming decades, scientists and mathematicians will come to see the false distinctions between pure and applied mathematics. ... More and more, mathematicians will see their subject as underlying all science and social science -- not as a humble servant but as an essential companion."
Bust to boom at Rochester. That's how Mark Clayton of the Christian Science Monitor (August 13 2002) describes the University of Rochester Math Department's fight back from the brink of losing its graduate program. How did they do it? They sold their undergraduates on mathematics to the point where they now have more than 5% of the majors. But the good news for Rochester points up the bleak picture nation-wide. " ... bachelor's degrees granted in mathematics fell 19 percent between 1990 and 2000, even though overall undergraduate enrollment rose 9 percent." Clayton examines various explanations for this phenomenon. Poor preparation of students in high school? Math's reputation as nerdy or too demanding? He describes NSF efforts over the last decade to remedy the problem by supporting research in math education and by providing scholarships for undergraduate summer research. At the end of the article Clayton comes back to Rochester. What was their secret? Part of it seems to have been "the inscrutable P-factor - the strange power of free pizza over the undergraduate."